The mathematical envelopes to families of both rigid and non-rigid shapes (or, equivalently, of moving shapes) are fundamental to a variety of problems from very diverse application domains, such as engineering design, computer aided manufacturing and path planning, problems characterized by evolving interfaces, geometric modeling or computer graphics. Singularities in these envelopes are known to induce malfunction or unintended system behavior, but the corresponding theoretical and computational difficulties are both massive and well documented. Despite their long history, the detection of these singularities remains one of the significant open problems faced in engineering today, which effectively precludes any attempts to develop systematic and generic approaches for eliminating these singularities.
Sweeping an object through space is one of the fundamental operations in geometric modeling. Furthermore, many practical problems from very diverse fields ranging from engineering design and manufacturing to robotics, computer graphics and computer assisted surgery can be formulated in terms of sweeps. The boundary of a set swept by an object in motion is expressed mathematically as the envelope to the family of shapes defined by the moving object. These envelopes, which are defined as solutions to specific differential equations, are known to exhibit geometric singularities. In practical terms, these envelope singularities induce malfunctions or unintended system behavior in the corresponding applications. However, the corresponding theoretical and computational difficulties induced by these singularities are not only massive, but also well documented.
Consider a simple cam-follower mechanism used to transform a rotation with constant angular velocity of a shaft into a relatively complex output motion, such as that illustrated in FIG. 1. Such a design problem is typically formulated in terms of the set swept by the moving follower relative to the cam. The boundary of such a cam is a subset of the envelopes of the moving follower (in this case a 2D disk). Though, for a given follower as well as cam and follower motions, the envelopes of the moving follower will not always be in contact with the follower. In other words, a cam that would move the follower according to the prescribed motion may not exist. This case is illustrated in FIG. 1, where the follower is moving relative to the cam such that its center moves along the specified trajectory. In this example, loss of contact between the follower and the resulting external cam would occur twice during a complete cycle of the relative motion.
The envelopes corresponding to the loss of contact have singular points, which is discussed in more detail below. This phenomenon is known as undercutting in the design of higher kinematic pairs, but it obviously occurs in all the application domains mentioned above. Once singularities, and therefore loss of contact, are detected, one can change either the motion, the geometry or a combination of these to correct the loss of contact. However, it is intuitively clear that the development of an algorithmic approach to eliminate this condition would require knowledge not only of whether singularities (and therefore undercutting) exists, but also of how severe this condition is. Furthermore, the knowledge of what part of the boundary of the moving object is responsible for generating these singularities during the given motion could lead to performing systematic geometric changes that eliminate this condition.
Unfortunately, the current approaches to detect these singularities are fairly specialized and scattered across all these application domains. The more advanced techniques require the ability to compute or approximate the envelopes, and most are applicable only to restricted classes of geometries and motions to an extent that some practical cases have no known solution. The fact that these approaches provide (at most) a binary answer, even for the classes of problems that they can handle, implies that the elimination of these singularities (e.g., via changes in geometry, motion or both) is forced to rely only on the accumulated engineering experience and heuristics. On the other hand, the ability to systematically detect, quantify, and eliminate these singularities, and hence the corresponding malfunctions, would be extremely beneficial to all these application domains. The reasons for the status quo discussed above are well documented: mathematical singularities are known to create extreme difficulties, not only in the analytical realm, but also in the computational domain.
Broadly, there are two categories of approaches for detecting geometric singularities in the envelopes of moving surfaces. The first category includes those (traditional) approaches that provide a closed form solution to the existence of these singularities. Unfortunately, such conditions can be derived only in those cases that place severe restrictions on the geometry and motion, which prevents them from being applicable to most “interesting” practical situations.
The second category of approaches rely on the theory on envelopes to detect these singularities, and, consequently, the associated system malfunctions. Though, these approaches are limited by the ability to compute or approximate the envelopes, which remains a difficult problem for general shapes and motions. Hence, these approaches are also forced to restrict the class of problems so that the envelopes can be approximated.
In the field of mechanical design, the loss of contact (or “undercutting”) in a higher kinematic pair has been known for over a century and represents one of the significant roadblocks in the design and manufacturing of higher pairs. Many of the earlier techniques for detecting loss of contact in the design of higher pairs fall in the first category mentioned above, and exploit the relatively simple geometry of the pairs (cylindrical or planar follower). Thus, these techniques can be applied only to the cases satisfying very specific and limiting geometric assumptions. The recent methods to design cam mechanisms have used envelope computations for cylindrical or flat faced followers. Almost independently from these efforts, differential geometry has been used in the gear design literature to develop mathematical conditions of non-existence for envelope singularities. The derivation of these conditions relies on a theorem relating the existence of the singular points to the zero valued sliding velocity at the contact point. All these techniques address specific classes of problems, and require difficult numerical computations, which, in turn, limits their applicability. Moreover, undercutting conditions for some classes of gearing mechanisms, such as wormgears, do not seem to be presently known.
The need to detect the singularities in the envelopes occurs in other applications as well. For example, envelope singularities are used in computer aided manufacturing and computer aided process planning to describe workpiece undercutting resulting in part defects. In this context, similar techniques based on the theory of envelopes have been coupled with assumptions on simple tool geometries to develop specific undercutting conditions. Singularities and their computational properties also play a significant role in sweep boundary evaluation; in Minkowski operations and offsets computations; in problems involving moving fronts, and in computing the geometry of shadows in computer graphics applications.
One of the recent techniques for computing offset curves and surfaces uses rational distance maps to eliminate the portions of the envelopes that are part of the set swept by a moving object. This approach is aimed at computing the boundary of the sweep, and does not quantify the regions where the loss of contact occurs. Recent attempts to detect the envelope singularities rely on envelope computations one way or another, and face the limitations discussed above. Importantly, none of these approaches quantifies the severity of these singularities, or of the associated malfunctions (such as the undercutting or overcutting in higher kinematic pairs), and give practically no support to the task of eliminating these singularities.